05A16-TakeuchiFunction.tex

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\pmtitle{Takeuchi function}
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The {\em Takeuchi function} is a triply recursive 3-parameter function originally defined by Ichiro Takeuchi in 1978 as $t(x, y, z) = y$ if $x \leq y$ and $t(x, y, z) = t(t(x - 1, y, z), t(y - 1, z, x), t(z - 1, x, y))$ otherwise. Later John McCarthy simplified the definition of the function as $t(x, y, z) = y$ if $x \leq y$, $t(x, y, z) = z$ if $y \leq z$ and $t(x, y, z) = x$ in all other cases.

For example, $t(194, 13, 5) = 194$ since 194 is not less than 13, and 13 is not less than 5. The return value of the function ``is on no practical significance,'' but the function itself ``is useful for benchmark testing of programming languages.'' (Finch, 2003) The function $T(x, y, z)$ is the number of times $t$ calls itself to obtain the return value. A properly optimized implementation of the function in a given programming language should not require more recursion than $T$ indicates.

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\bibitem{sf} Steven R. Finch {\it Mathematical Constants} New York: Cambridge University Press (2003): 321
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